STOCHASTIC MODELLING OF STRUCTURED POPULATIONS

量化细胞生长对于了解微生物和癌细胞等细胞群体的动态至关重要。分批培养的标准行为是众所周知的，它的特征通常是在开始指数生长之前的延迟、指数阶段和稳定阶段；然而，在单细胞水平上，由于细胞周期持续时间的波动、环境变化引起的可变性以及细胞之间的相互作用，细胞间的生长差异很大。目前，理解细胞间的可变性如何影响整个细胞的进化仍然是一个开放的挑战；事实上，仍然缺乏可靠的理论和模拟方法来预测细胞异质性对种群动态的影响[1]。我们提出了一个新的随机模型，其中细胞由分裂、死亡、转换到其他物种、响应于随时间增加的内部连续状态而恢复活力的个体来表示。虽然这样的模型通常只适用于模拟，但我们证明了种群结构可以用一个泛函主方程来描述，这个主方程可以被操纵来获得一个新的积分更新方程。与Bellman Harris分枝过程[2]和Galton-Watson理论[3]中关于更新理论的经典结果相比，后者更进一步。实际上，它为细胞异质性对种群动态的作用提供了坚实而紧凑的随机描述，该分析框架使我们能够完整地描述种群规模分布、种群增长率、祖先和分裂时间分布，也使我们能够理解异质性在初始条件下所起的作用。此外，我们还给出了任意细胞间异质性范围内的灭绝概率和首次灭绝时间分布的解析和数值表征。我们还提出了一种新的方法来模拟细胞群体的进化受个体变异性的影响。这样的计算工具使我们能够证实在这项研究中获得的分析和数值结果。我们的最新结果也提供了新的方法来解决细胞到细胞变异性在时间依赖环境中的作用。结果表明，在反应网络速率除细胞内部特性外，反应网络速率明显依赖于时间的情况下，基于智能体的群体行为也可以得到随机描述。我们的结果为理解各种生物学和医学应用中潜在的波动的随机机制开辟了一条新的理论途径：正在治疗的癌细胞种群的灭绝，不利环境中的细胞种群增长，乳腺癌的休眠-觉醒转变和微生物的静止。参考[1]Thomas Philipp(2017)。R.Soc.142017046720170467 http://doi.org/10.1098/rsif.2017.0467[2]Harris，T.E.(1963年)。《分支过程理论》(第6卷)。[3]Kesten，H.Ney，P.，Spitzer，F.(1966)。具有均值1和有限方差的Galton-Watson过程。概率论及其应用。工业和应用数学学会。10.1137/1111059 https：//doi.org/10.1137/1111059

Quantifying cellular growth is crucial to understand the dynamics of cell populations such as microbes and cancer cells. The standard behaviour of batch cultures is well known and it is usually characterised by a delay before the start of exponential growth, an exponential phase, and a steady phase; however, at the single-cell level, growth varies drastically from cell to cell due to the fluctuations in the cell cycle duration, variability caused by changing environments, and cells interactions.At the present time, understanding how the cell-to-cell variability affects the evolution of the entire popu-lation is still a open challenge; de facto, there are still lacking solid theoretical and simulation methods to forecast the effects of cell heterogeneity on the population dynamics [1].We propose a novel stochastic model where the cells are represented by agents who divide, die, convert to other species, rejuvenate in response to an internal continuous state which increases with time. While such models are usually only amenable to simulations, we show that the population structure can be characterized by a functional master equation which can be manipulated to obtain a novel integral renewal equation. Compared to the classic results about renewal theory, as the Bellman Harris branching process [2] and the Galton-Watson theory [3], the latter equation takes a step further. In fact, it provides a solid and compact stochastic description of the role played by cell heterogeneity on the population dynamics.The analytical framework allowed us to fully describe the population size distribution, population growth rate, ancestor and division times distributions; it also enables to understand the role played by heterogeneity in the initial conditions. Moreover, we provide an analytical and numerical characterization of the extinction probability and first extinction times distribution for any cell-to-cell heterogeneity range. We also propose a novel way to simulate the evolution of cell populations affected by the variability of the individuals. Such computational tool allowed us to substantiate the analytical and numerical results obtained during this in-vestigation.Our last results also provide novel methods to address the role of cell-to-cell variability in time depen-dent environments. We showed that the stochastic description of agent-based populations dynamics can be obtained in scenario where the reaction network rates depend explicitly on time in addition to the internal traits of the cells.In conclusion, the following research project proposes a novel methodology to describe the stochastic be-haviour of cell structured population with numerical, computational and analytical methods. Our results open a new theoretical path to understanding stochastic mechanisms underlying fluctuations in various bi-ological and medical applications as: extinction of cancer cell populations under treatment, cell population growth in adverse environments, dormancy-awakening transition in breast cancer and microbial quiescence.References[1] Thomas Philipp (2017).Making sense of snapshot data: ergodic principle for clonal cell populationsJ. R. Soc. Interface.142017046720170467 http://doi.org/10.1098/rsif.2017.0467[2] Harris, T. E. (1963). The theory of branching processes (Vol. 6). Berlin: Springer.[3] Kesten, H.Ney, P., Spitzer, F. (1966). The Galton-Watson Process with Mean One and Finite Variance. Theory of Probability Its Applications. Society for Industrial and Applied Mathematics. 10.1137/1111059 https: //doi.org/10.1137/1111059